The development of delay-sensitive applications that require ultra high reliability created an additional challenge for wireless networks. This led to Ultra-Reliable Low-Latency Communications, as a use case that 5G and beyond 5G systems must support. However, supporting low latency communications requires the use of short codes, while attaining vanishing frame error probability (FEP) requires long codes. Thus, developing codes for the finite blocklength regime (FBR) achieving certain reliability requirements is necessary. This paper investigates the potential of Convolutional Neural Networks autoencoders (CNN-AE) in approaching the theoretical maximum achievable rate over a Gaussian channel for a range of signal-to-noise ratios at a fixed blocklength and target FEP, which is a different perspective compared to existing works that explore the use of CNNs from bit-error and symbol-error rate perspectives. We explain the studied CNN-AE architecture, evaluate it numerically, and compare it to the theoretical maximum achievable rate and the achievable rates of polar coded quadrature amplitude modulation (QAM), Reed-Muller coded QAM, multilevel polar coded modulation, and a TurboAE-MOD scheme from the literature. Numerical results show that the CNN-AE outperforms these benchmark schemes and approaches the theoretical maximum rate, demonstrating the capability of CNN-AEs in learning good codes for delay-constrained applications.